2 .1 Basic Concepts 2 .2 Frequent Itemset Mining Methods 2 .3 Which Patterns Are Interesting? Pattern Evaluation M

Chapter 2: Mining Frequent Patterns, Associations and Correlations • 2.1 Basic Concepts • 2.2 Frequent Itemset Mining Methods • 2.3 Which Patterns Are Interesting? • Pattern Evaluation Methods • 2.4 Summary

Frequent Pattern Analysis • Frequent Pattern: a pattern (a set of items, subsequences, substructures, etc.) that occurs frequently in a data set • Goal: finding inherent regularities in data • What products were often purchased together?— Beer and diapers?! • What are the subsequent purchases after buying a PC? • What kinds of DNA are sensitive to this new drug? • Can we automatically classify Web documents? • Applications: • Basket data analysis, cross-marketing, catalog design, sale campaign analysis, Web log (click stream) analysis, and DNA sequence analysis.

Why is Frequent Pattern Mining Important? • An important property of datasets • Foundation for many essential data mining tasks • Association, correlation, and causality analysis • Sequential, structural (e.g., sub-graph) patterns • Pattern analysis in spatiotemporal, multimedia, time-series, and stream data • Classification: discriminative, frequent pattern analysis • Clustering analysis: frequent pattern-based clustering • Data warehousing: iceberg cube and cube-gradient • Semantic data compression • Broad applications

Frequent Patterns • itemset: A set of one or more items • K-itemset X = {x1, …, xk} • (absolute) support, or, support count of X: Frequency or occurrence of an itemset X • (relative) support, s, is the fraction of transactions that contains X (i.e., the probability that a transaction contains X) • An itemset X is frequent if X’s support is no less than a minsup threshold Customer buys both Customer buys diaper Customer buys beer

Association Rules • Find all the rules X  Ywith minimum support and confidence threshold • support, s, probability that atransaction contains X  Y • confidence, c, conditional probability that a transaction having X also contains Y Let minsup = 50%, minconf = 50% Freq. Pat.: Beer:3, Nuts:3, Diaper:4, Eggs:3, {Beer, Diaper}:3 • Association rules: (many more!) • Beer  Diaper (60%, 100%) • Diaper  Beer (60%, 75%) • Rules that satisfy both minsup and minconf are called strong rules Customer buys both Customer buys diaper Customer buys beer

Closed Patterns and Max-Patterns • A long pattern contains a combinatorial number of sub-patterns, e.g., {a1, …, a100}contains (1001) + (1002) + … + (110000) = 2100 – 1 = 1.27*1030 sub-patterns! • Solution: Mine closed patterns and max-patterns instead • An itemset Xis closedif X is frequent and there exists no super-pattern Y כ X, with the same support as X • An itemset X is a max-pattern if X is frequent and there exists no frequent super-pattern Y כ X • Closed pattern is a lossless compression of freq. patterns • Reducing the number of patterns and rules

Closed Patterns and Max-Patterns Example • DB = {<a1, …, a100>, < a1, …, a50>} • Min_sup=1 • What is the set of closed itemset? • <a1, …, a100>: 1 • < a1, …, a50>: 2 • What is the set of max-pattern? • <a1, …, a100>: 1 • What is the set of all patterns? • !!

Computational Complexity • How many itemsets are potentially to be generated in the worst case? • The number of frequent itemsets to be generated is sensitive to the minsup threshold • When minsup is low, there exist potentially an exponential number of frequent itemsets • The worst case: MN where M: # distinct items, and N: max length of transactions

Chapter 2: Mining Frequent Patterns, Associations and Correlations • 2.1 Basic Concepts • 2.2 Frequent Itemset Mining Methods 2.2.1 Apriori: A Candidate Generation-and-Test Approach 2.2.2 Improving the Efficiency of Apriori 2.2.3 FPGrowth: A Frequent Pattern-Growth Approach 2.2.4 ECLAT: Frequent Pattern Mining with Vertical Data Format • 2.3 Which Patterns Are Interesting? • Pattern Evaluation Methods • 2.4 Summary

2.2.1 Apriori: Concepts and Principle • The downward closure property of frequent patterns • Any subset of a frequent itemset must be frequent • If {beer, diaper, nuts} is frequent, so is {beer, diaper} • i.e., every transaction having {beer, diaper, nuts} also contains {beer, diaper} • Apriori pruning principle:If there isanyitemset which is infrequent, its superset should not be generated/tested

Apriori Principle • Initially, scan DB once to get frequent 1-itemset • Generate length (k+1) candidate itemsets from length k frequent itemsets • Test the candidates against DB • Terminate when no frequent or candidate set can be generated

Apriori: Example Supmin = 2 Database L1 C1 1st scan C2 C2 L2 2nd scan L3 C3 3rd scan

Apriori Algorithm Ck: Candidate itemset of size k Lk : frequent itemset of size k L1 = {frequent items}; for(k = 1; Lk !=; k++) do begin Ck+1 = candidates generated from Lk; for each transaction t in database do increment the count of all candidates in Ck+1 that are contained in t Lk+1 = candidates in Ck+1 with min_support end returnkLk;

Candidate Generation • How to generate candidates? • Step 1: self-joining Lk • Step 2: pruning Join Lkp with Lkq, as follows: insertinto Ck+1 select {p.itemi}{1,..,k-1}, p.itemk, q.itemk fromLkp, Lkq where{p.itemi}{1,..k-1} = {q.itemi} {1,..k-1} and p.itemk< q.itemk

Example of Candidate Generation Suppose we have the following frequent 3-itemsets and we would like to generate the 4-itemsets candidates • L3={{I1, I2, I3} , {I1, I2, I4}, {I1, I3, I4}, {I1, I3, I5}, {I2,I3,I4}} • Self-joining: L3*L3 gives: {I1,I2,I3,I4} from {I1, I2, I3} , {I1, I2, I4}, and {I2,I3,I4} {I1,I3,I4,I5} from {I1, I3, I4} and {I1, I3, I5} • Pruning: {I1,I3,I4,I5} is removed because {I1,I4,I5} is not in L3 • C4 = {I1,I2,I3,I4}

Generating Association Rules • Once the frequent itemsets have been found, it is straightforward to generate strong association rules that satisfy: • minimum support • minimum confidence • Relation between support and confidence: support_count(AB) Confidence(AB) = P(B|A)= support_count(A) • Support_count(AB) is the number of transactions containing the itemsets A  B • Support_count(A) is the number of transactions containing the itemset A.

Generating Association Rules • For each frequent itemsetL, generate all non empty subsets of L • For every non empty subsetSofL, output the rule: • S  (L-S) If (support_count(L)/support_count(S)) >= min_conf

Example • Suppose the frequent Itemset L={I1,I2,I5} • Subsets of L are: {I1,I2}, {I1,I5},{I2,I5},{I1},{I2},{I5} • Association rules : I1  I2  I5 confidence = 2/4= 50% I1  I5  I2 confidence=2/2=100% I2  I5  I1 confidence=2/2=100% I1  I2  I5 confidence=2/6=33% I2  I1  I5 confidence=2/7=29% I5  I2  I2 confidence=2/2=100% If the minimum confidence =70% Question: what is the difference between association rules and decision tree rules? Transactional Database

2.2.2 Improving the Efficiency of Apriori • Major computational challenges • Huge number of candidates • Multiple scans of transaction database • Tedious workload of support counting for candidates • Improving Apriori: general ideas • Shrink number of candidates • Reduce passes of transaction database scans • Facilitate support counting of candidates

(A) DHP: Hash-based Technique Database C1 1st scan Making a hash table 10: {A,C}, {A, D}, {C,D} 20: {B,C}, {B, E}, {C,E}30: {A,B}, {A, C}, {A,E}, {B,C}, {B, E}, {C,E}40: {B, E} Hash codes Buckets Buckets counters 1 0 1 0 1 0 1 {A,B} 1 {A, C} 2 {B,C} 2 {B, E} 3 {C,E} 2 min-support=2 We have the following binary vector {A, C} {B,C} {B, E} {C,E} J. Park, M. Chen, and P. Yu. An effective hash-based algorithm for mining association rules. SIGMOD’95

(A) DHP: Hash-based Technique Database C1 1st scan At the same time of counting the support of Itemet-1, generate itemset2 Make a hash table using: h({x,y})= ((order of x)*10 + (order of y)) mod 7 Hash codes Buckets Buckets counters 1 0 1 0 1 0 1 {I1,I2} 1 {I1, I3} 3 L1 ×L1 {I1,I5} 1 {I2,I3} 2 {I2,I5} 3 {I3,I5} 3 min-support=2 We have the following binary vector {I1, I3} {I2,I3} {I2, I5} {I3,I5} J. Park, M. Chen, and P. Yu. An effective hash-based algorithm for mining association rules. SIGMOD’95

(B) Partition: Scan Database Only Twice • Subdivide the transactions of D into k non overlapping partitions • Any itemset that is potentially frequent in D must be frequent in at least one of the partitions Di • Each partition can fit into main memory, thus it is read only once • Steps: • Scan1: partition database and find local frequent patterns • Scan2: consolidate global frequent patterns D1 + D2 + + Dk = D • A. Savasere, E. Omiecinski and S. Navathe, VLDB’95

(C) Sampling for Frequent Patterns • Select a sample of the original database • Mine frequent patterns within the sample using Apriori • Use a lower support threshold than the minimum support to find local frequent itemsets • Scan the database once to verify the frequent itemsets found in the sample • Only broader frequent patterns are checked • Example: check abcd instead of ab, ac,…, etc. • Scan the database again to find missed frequent patterns H. Toivonen. Sampling large databases for association rules. In VLDB’96

(D) Dynamic: Reduce Number of Scans • Once both A and D are determined frequent, the counting of AD begins • Once all length-2 subsets of BCD are determined frequent, the counting of BCD begins ABCD ABC ABD ACD BCD Transactions AB AC BC AD BD CD 1-itemsets 2-itemsets Apriori … B C D A 1-itemsets {} 2-items Itemset lattice DIC 3-items • S. Brin R. Motwani, J. Ullman, and S. Tsur. Dynamic itemset counting and implication rules for market basket data. In SIGMOD’97

2.2.3 FP-growth: Frequent Pattern-Growth • Adopts a divide and conquer strategy • Compress the database representing frequent items into a frequent –pattern tree or FP-tree • Retains the itemset association information • Divid the compressed database into a set of conditional databases, each associated with one frequent item • Mine each such databases separately

Example: FP-growth • The first scan of data is the same as Apriori • Derive the set of frequent 1-itemsets • Let min-sup=2 • Generate a set of ordered items Transactional Database

Construct the FP-Tree Transactional Database - Create a branch for each transaction - Items in each transaction are processed in order 1- Order the items T100: {I2,I1,I5} 2- Construct the first branch: <I2:1>, <I1:1>,<I5:1> null I2:1 I1:1 I5:1

Construct the FP-Tree Transactional Database - Create a branch for each transaction - Items in each transaction are processed in order 1- Order the items T200: {I2,I4} 2- Construct the second branch: <I2:1>, <I4:1> null I2:2 I2:1 I4:1 I1:1 I5:1

Construct the FP-Tree Transactional Database - Create a branch for each transaction - Items in each transaction are processed in order 1- Order the items T300: {I2,I3} 2- Construct the third branch: <I2:2>, <I3:1> null I2:2 I2:3 I4:1 I1:1 I3:1 I5:1

Construct the FP-Tree Transactional Database - Create a branch for each transaction - Items in each transaction are processed in order 1- Order the items T400: {I2,I1,I4} 2- Construct the fourth branch: <I2:3>, <I1:1>,<I4:1> null I2:4 I2:3 I4:1 I1:2 I1:1 I3:1 I5:1 I4:1

Construct the FP-Tree Transactional Database - Create a branch for each transaction - Items in each transaction are processed in order 1- Order the items T400: {I1,I3} 2- Construct the fifth branch: <I1:1>, <I3:1> null I1:1 I2:4 I4:1 I1:2 I3:1 I3:1 I5:1 I4:1

Construct the FP-Tree Transactional Database null I1:2 I2:7 I4:1 I1:4 I3:2 I3:2 I5:1 I4:1 When a branch of a transaction is added, the count for each node along a common prefix is incremented by 1 I3:2 I5:1

Construct the FP-Tree null I2:7 I1:2 I1:4 I4:1 I3:2 I3:2 I5:1 I4:1 I3:2 I5:1 The problem of mining frequent patterns in databases is transformed to that of mining the FP-tree

Construct the FP-Tree null I2:7 I1:2 I1:4 I4:1 I3:2 I3:2 I5:1 I4:1 I3:2 I5:1 -Occurrences of I5:<I2,I1,I5> and <I2,I1,I3,I5> -Two prefix Paths<I2, I1: 1> and <I2,I1,I3: 1> -Conditional FP tree contains only <I2: 2, I1: 2>, I3 is not considered because its support count of 1 is less than the minimum support count. -Frequent patterns {I2,I5:2}, {I1,I5:2},{I2,I1,I5:2}

Construct the FP-Tree null I2:7 I1:2 Item ID Support count I1:4 I2 7 I4:1 I3:2 I1 6 I3:2 I3 6 I5:1 I4 2 I4:1 I5 2 I3:2 I5:1

FP-growth properties • FP-growth transforms the problem of finding long frequent patterns to searching for shorter once recursively and concatenating the suffix • It uses the least frequent suffix offering a good selectivity • It reduces the search cost • If the tree does not fit into main memory, partition the database • Efficient and scalable for mining both long and short frequent patterns

2.2.4 ECLAT: FP Mining with Vertical Data Format • Both Apriori and FP-growth use horizontal data format • Alternatively data can also be represented in vertical format

ECLAT Algorithm by Example • Transform the horizontally formatted data to the vertical format by scanning the database once • The support count of an itemset is simply the length of the TID_set of the itemset

ECLAT Algorithm by Example Frequent 1-itemsets in vertical format • The frequent k-itemsets can be used to construct the candidate (k+1)-itemsets based on the Apriori property min_sup=2 Frequent 2-itemsets in vertical format

ECLAT Algorithm by Example Frequent 3-itemsets in vertical format • This process repeats, with k incremented by 1 each time, until no frequent items or no candidate itemsets can be found • Properties of mining with vertical data format • Take the advantage of the Apriori property in the generation of candidate (k+1)-itemset from k-itemsets • No need to scan the database to find the support of (k+1) itemsets, for k>=1 • The TID_set of each k-itemset carries the complete information required for counting such support • The TID-sets can be quite long, hence expensive to manipulate • Use diffsettechnique to optimize the support count computation min_sup=2

Strong Rules Are Not Necessarily Interesting • Whether a rule is interesting or not can be assessed either subjectively or objectively • Objective interestingness measures can be used as one step toward the goal of finding interesting rules for the user • Example of a misleading “strong” association rule • Analyze transactions of AllElectronics data about computer games and videos • Of the 10,000 transactions analyzed • 6,000 of the transactions include computer games • 7,500 of the transactions include videos • 4,000 of the transactions include both • Suppose that min_sup=30% and min_confidence=60% • The following association rule is discovered: Buys(X, “computer games”)  buys(X, “videos”)[support =40%, confidence=66%]

Strong Rules Are Not Necessarily Interesting Buys(X, “computer games”)  buys(X, “videos”)[support 40%, confidence=66%] • This rule is strong but it is misleading • The probability of purshasing videos is 75% which is even larger than 66% • In fact computer games and videos are negatively associated because the purchase of one of these items actually decreases the likelihood of purchasing the other • The confidence of a rule A  B can be deceiving • It is only an estimate of the conditional probability of itemset B given itemset A. • It does not measure the real strength of the correlation implication between A and B • Need to use Correlation Analysis

From Association to Correlation Analysis • Use Lift, a simple correlation measure • The occurrence of itemset A is independent of the occurrence of itemset B if P(AB)=P(A)P(B), otherwise itemsets A and B are dependent and correlated as events • The lift between the occurences of A and B is given by Lift(A,B)=P(AB)/P(A)P(B) • If > 1, then A and B are positively correlated (the occurrence of one implies the occurrence of the other) • If <1, then A and B are negatively correlated • If =1, then A and B are independent • Example: P({game, video})=0.4/(0.60  0.75)=0.89

2.4 Summary • Basic Concepts: association rules, support-confident framework, closed and max patterns • Scalable frequent pattern mining methods • Apriori (Candidate generation & test) • Projection-based (FPgrowth) • Vertical format approach (ECLAT) • Interesting Patterns • Correlation analysis

2 .1 Basic Concepts 2 .2 Frequent Itemset Mining Methods 2 .3 Which Patterns Are Interesting? Pattern Evaluation M